Why does the Möbius function take its values so often in $\{0,+1,-1\}$?

Question

The Möbius function of a locally finite poset $P$ is defined on its intervals $[x,y] \subseteq P$ recursively by $$\mu([x,x])=1$$ $$\forall x < y : \mu([x,y])=-\sum_{x \leq z < y} \mu([x,z])$$ In many examples (see Wikipedia) it turns out that $\mu$ takes its values in $\{0,+1,-1\}$. Is there any conceptual reason for this? Also, is there some characterization of those posets $P$ for which this holds?

Answer 1

Every one of the ones in that list which only take values $1,-1,0$ has the property that every range $[a,b]$ is isomorphic to some range in $\mathbb N^k$, the product of $k$ copies of the poset $\mathbb N$.

We can show that the Möbius function for $P_1\times P_2$ is the product:

$$\mu_{P_1\times P_2}([(a_1,a_2),(b_1,b_2)]) = \mu_{P_1}([a_1,b_1])\mu_{P_2}([a_2,b_2])$$

So if you know that the Möbius function on $\mathbb N$ only takes values $0,1,-1$, you get that the Möbius functions for finite sets, finite multi-sets, natural numbers under divisibility, can only take values $0,1,-1$.

One key realization is that $\mu([a,b])$ is entirely determined by the isomorphism class of the sub-poset $[a,b]=\{x\in P\mid a\leq x\leq b\}$. Two intervals which are isomorphic as posets will be have the same value for the Möbius function, even if the intervals are in different posets.

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Another view is that all of these posets are isomorphic to filters of the Multisets poset. So the values taken by each can only be the values take by Multisets.